A Polynomial-Time Algorithm for the Next-to-Shortest Path Problem on Positively Weighted Directed Graphs

TL;DR

This paper presents a polynomial-time algorithm for the next-to-shortest path problem in directed graphs with positive edge weights, resolving a long-standing open problem in graph theory and algorithms.

Contribution

It introduces the first polynomial-time algorithm for the next-to-shortest path problem on directed graphs with positive weights, previously unresolved.

Findings

Algorithm successfully computes next-to-shortest paths in directed graphs with positive weights.

The problem is shown to be solvable in polynomial time, unlike the NP-complete case for non-negative weights.

The work closes a 30-year gap in the understanding of this problem.

Abstract

Given a graph and a pair of terminals $s$, $t$, the next-to-shortest path problem asks for an $s\!\to \!t$ (simple) path that is shortest among all not shortest $s\!\to \!t$ paths (if one exists). This problem was introduced in 1996, and soon after was shown to be NP-complete for directed graphs with non-negative edge weights, leaving open the case of positive edge weights. Subsequent work investigated this open question, and developed polynomial-time algorithms for the cases of undirected graphs and planar directed graphs. In this work, we resolve this nearly 30-year-old open problem by providing an algorithm for the next-to-shortest path problem on directed graphs with positive edge weights.